The unscented Kalman filter has been increasingly popular in engineering applications since its introduction about twenty years ago [1]}, [2]}. In a UKF algorithm, the error covariance is propagated with the dynamics using a set of vectors, or \(\sigma \) -points denoted by \(x^\sigma \) . Their definition is given in (REF )-(REF ). The \(\sigma \) -points are computed at each time step using a square root of the error covariance. In most UKF applications, \(\sigma \) -points are computed using either Cholesky factorization or matrix diagonalizations. In the notation, a variable with a superscript '\(a\) ', such as \(x^a\) , represents the analysis value of the variable, i.e., the updated value based on observations. A variable with a superscript '\(b\) ', such as \(y^b\) , represents the background, i.e., the propagated value of analysis using the system model. The algorithm is summarized as follows. At \(t=k-1\) , suppose we have the analysis and error covariance as well as its square root
\(\begin{array}{lllllllll}x^a (k-1), \; P^a(k-1),\\X^a(k-1)=\sqrt{(n+\kappa )P^a(k-1)},\end{array}\) 
